Using Rough Sets for Optimal Cost Evaluation in Supply Chain Management Shobhit Tiwari
Sanjiban Sekhar Roy
School of Computer Science and Engineering Vellore Institute of Technology Vellore, India
[email protected]
School of Computer Science and Engineering Vellore Institute of Technology Vellore, India
[email protected]
Abstract—Over the last decade we have seen rapid advancement in the field of rough set theory. It has been successfully been applied to many varied fields such as data mining and network intrusion detection with little or no modifications. The concept of rough set is increasingly becoming popular which can be easily seen from the increasing number of research articles devoted to it. In the past several studies have targeted on finding the qualitative causal relationships that exist between various businesses and their associated attributes. This has resulted in the need for a quantitative approach for the evaluation of cost for supply chain management. However since the process for supply chain management itself depends on multiple complicated features regarding which the standard statistical techniques are not deemed suitable. Therefore in this paper we have analyzed and demonstrated a method for optimal cost evaluation using the rough set theory.
Keywords-supply chain management; rough sets; cost evaluation I.
INTRODUCTION
The change in global macro and micro economic scenario has prompted companies to give more importance to supply chain management. Efficient management of supply chain is important to maintain profitability and business competitiveness. Most business decisions are guided by financial profitability constraints, similarly in the case of supply chain management cost evaluation is a very essential factor to ensure sustainable supply chain management. Cost considerations are more important in case of SME’s which operate on a limited budget. In such cases the profitability of the enterprise greatly depends on analyzing financial aspects of different supply chain management and outsourcing strategies. In this paper we have shed some light on the concepts of rough set theory and later in the paper we have shown how rough set theory can be utilized for cost evaluation in supply chain management. “Fig. 1” shows the conceptual framework of a supply chain for regular business enterprise. II.
LITERATURE REVIEW
Mentzer et al.[3] defined supply chain management as, ‘‘the systemic, strategic coordination of the traditional business functions and the tactics across these business
Madhu Viswanatham V School of Computer Science and Engineering Vellore Institute of Technology Vellore, India
[email protected] Supplier
Manufacturer
Distributor
Customer Figure 1. Conceptual Framework of a Supply Chain. (The dashed line indicates manufacturers who sell directly to final consumers)
functions within a particular company and across businesses within the supply chain, for the purposes of improving the long-term performance of the individual companies and the supply chain as a whole’’. In their work Lambert et al. [4] proposed that it refers to ‘‘the integration of key business processes from end-user through original suppliers, that provides products, services, and information that add value for customers and other stakeholders’’. Lindqvist [5] in his work reviewed the research trends in distribution in Finland and discovered that the attributes influencing the length of the distribution channel, the variables accounting for size of retail trade in commune level centres, and the effect of the location and dimension of the automobile dealership on its profitability are at the major factors to be considered in distribution research. While reviewing the existing literature on supply chain management we find that most of them have employed qualitative techniques to examine the causal relationships in supply chain management. These qualitative studies have identified insightful managerial implications for supply chain management. However quantitative methods which provide a more accurate and precise solution to the decision based problem are being developed. While developing algorithms for diagnosing supply chain management, the various
attributes of uncertainty and their impact on the supply chain should be carefully evaluated. III.
B ( X ) = {B ( x ) : B ( x ) ⊆ X }
(2)
B ( X ) = {B ( x ) : B ( x ) ∩ X ≠ φ }
(3)
x∈U
ROUGH SET THEORY
The theory of Rough set is a new mathematical tool to deal with intelligent data analysis and data mining [1] proposed by Z Pawlak. There are much developments happened in the recent years with the rough set theory. The theory of rough set based on the concept that every object in the universe is attached with some kind of information. Set theory is a great help to the computer science research and theory of Rough set is an extension to that. It’s a mathematical tool to deal with inexact, uncertain or vague data, which are part of artificial intelligent system. Rough set also includes algorithms for generation of rules, classification and reduction of attributes[2]. It is hugely used for knowledge discovery and reduction of knowledge. The theory of Rough set has got many important applications. Without using statistical or probabilistic approach, the rough set theory can do data analysis, which is why it has got so much popularity and importance. Greco et al.[6] extended the original version of Rough Set Theory in a number of directions in order to deal with problems of multi-criteria decision analysis (MCDA). Daubie et al.[7] compared the rough set and decision tree approaches as techniques for classifying credit applicants. Mickee[8] applied Rough Set Theory to deal with the problem of apparent indiscemibility between objects in a set. Wei et al.[9] combined the fuzzy set and rough set. Kumar et al.[10]explored the use of rough-set methods for marketing decision support systems in the retail business. A. Information Systems First, The rough sets theory starts with information represented by atable called an information system (Pawlak 1991). An information system is a 4-tuple S = (U, A, Va, fa), where: 1) U is the universe, a nonempty finite set of objects. 2) A ={a1,a2 ,...,am} is a nonempty finite set of attributes C ы D , where C and D is a finite set of condition and decision attributes, respectively. 3) Va is a domain of the attribute a, each attribute a :U ൺVa for
x∈U
That is, the elements of B(X ) are all the elementary objects certainly belonging to X. The elements of B(X ) are at least one object belonging to X. With the lower and upper approximation of a set X ๙U , the universe can be divided into three regions, the boundary region BND(X ) , the positive region POS(X ) , and the negative region NEG(X):
BND ( X ) = B( X ) − B ( X )
(4)
POS ( X ) = B ( X )
(5)
NEG ( X ) = U − B ( X )
(6)
If the boundary region of X is an empty set, BNB (X )
˳, then
X is a crisp set with respect to B; otherwise, if BNB (X ) ำ˳ , then X is a rough (approximate) set with respect to B. C. Set Approximation An equivalence relation induces a partitioning of the universe. These partitions can be used to build new subsets of the universe. Subsets that are most often of interest have the same value of the outcome attribute. It may happen, however, that a concept such as “Walk" cannot be defined in a crisp manner. These notions are formally expressed as follows. Let A = (U, A) be an information system and let B ⊆ A and
X ⊆ U . We can approximate X using only the information contained in B by constructing the B-lower and B-upper
approximations of X, denoted BX and BX respectively, where BX = { x | [ x ]B ⊆ X } and
(7)
(8) BX = {x | [ x ]B X ≠ φ } . The objects in BX can be with certainty classified as members of X on the basis of knowledge in B, while the
aෛA.
objects in BX can be only classified as possible members of X on the basis of knowledge in B. The set
4) fa :U A ൺ Va is the total decision function called the
BNB(X) = BX ⇔ BX is called the B-boundary region of
information function such that f (x,a)ෛVa for ීaෛA,ීxෛU .
X, and thus consists of those objects that we cannot decisively classify into X on the basis of knowledge in B. The set
B. Indiscernibility Relation Let S = (U, A, Va, fa) be an information system, B ๙A and X ๙ U . With any subset of attributes B ๙ A, a binary indiscernibility relation, is called B-indiscernibility relation, which is defined by: IND(B) {(x, y)ෛU U : a(x) a( y),ීaෛB}
(1)
For any subset X ๙U , the lower and upper approximation can be expressed as (5) and (6) respectively:
U ⇔ BX is called the B-outside region of X and consists of those objects which can be with certainty classified as do not belonging to X (on the basis of knowledge in B). A set is said to be rough (respectively crisp) if the boundary region is nonempty. D. Reduct and Core The other dimension in reduction is to keep only those attributes that preserve the indiscernibility relation and, consequently, set approximation. The rejected attributes are
redundant since their removal cannot worsen the classification. There is usually several such subsets of attributes and those which are minimal are called reducts. The set of attributes R ⊆ C is called a reduct of C, if T’ = (U, R, D) is independent and (9) POSR ( D ) = POSC ( D ) , The set of all the condition attributes indispensable in T is denoted by CORE(C). (10) CORE (C ) = RED (C ) , where RED(C) is the set of all reducts of C. E. Rough Membership The rough membership function quantifies the degree of relative overlap between the set X and the equivalence [ x ]B class to which x belongs. It is defined as follows
µ BX : U ⇔ [0,1] and µ BX ( x ) =
| [ x ]B X | | [ x ]B |
(11) The rough membership function can be interpreted as a frequency-based estimate of Pr( x ⊂ X | u ) , the conditional probability that object x belongs to set X, given knowledge u of the information signature of x with respect to attributes B. The formulae for the lower and upper set approximations can be generalized to some arbitrary level of precision
1 2
π ∈ ( ,1] , by means of the rough membership function, as shown below.
Bπ X = {x | μ XB ( x) ≥ π } (12) Note that the lower and upper approximations as originally formulated are obtained as a special case with ʌ = 1.0 F. Dependency of Attributes Another important issue in data analysis is discovering dependencies between attributes. Intuitively, a set of attributes D depends totally on a set of attributes C, denoted C D if all values of attributes from D are uniquely determined by values of attributes from C. Formally dependency can be defined in the following way. Let D and C be subsets of A. We will say that D depends on C in a degree k; (0 ≤ k ≤ 1) ,
C kD , if k = γ (C , D ) =
Where
POSC ( D ) =
| POSC ( D ) | |U |
(13)
n
X ∈U / D
called a positive region of the partition U/D with respect to C, is the set of all elements of U that can be uniquely classified to blocks of the partition U/D, by means of C. IV.
Bπ X = {x | μ XB ( x) > 1 − π }.
denoted
Figure 2. Conceptual Framework for taking decision rules
C( X ) (14)
PROPOSED MODEL
The information system would be coded as utilizing “High“ to represent that the item xi had higher average unit cost with respect to the attribute aj , “Medium” would have the fair average unit cost, and “Low” would be the low average unit cost. The indiscernibility relation table would be constructed to build up the information system. Let us consider the case of an enterprise with local supply chain, the universe set elements would be relative combinations of all members, and the attribute set for cost evaluations can have the following parameter distribution 1. average unit labour cost( a1 ), 2. average unit transportation cost( a 2 ), 3. average unit raw material cost( a 3 ), 4. average unit loss and other cost( a 4 ). Decision rules would be designed as per the requirements of the customers. For instance, if a certain customer needs medium-quality raw materials, low labour cost, and medium transportation costs then, the expecting decision vector for the given customer can be represented as D = {Low, Medium, Medium, Free} . “Fig. 2” shows a case of a parent company with various combinations of supply chain members.
Figure 3.
Parent company with various of supply chain members
The universe set would contain 6 temporary formed supply chains as shown in (15), and each combinations would be evaluated by four attribute properties as shown in (16) (15) U = {x1, x 2, x 3..., x 6}
A = {a1, a 2, a 3, a 4}
(16)
b1᧶ Payment delay, b2᧶Cost control, b3᧶Technical Skill, b4 ᧶ Infrastructure and equipment. According to the core business within the last six months to one year time, and the composition of the regional supply chain with the analysis of the situation, can be drawn as shown in Table I. To differentiate the identification ability between the members of each combination of local supply chain, by comparing TABLE I.
elements of the supply chain two-by-two we can obtain Boolean functions which by a combination of further rate can be simplified to obtain identification function. The simplified identification and estimated decision making cost is shown in Table II. In the current case the orders are dependent on the customer’s requirements. If the customers requires the relative value of raw materials cost ( a 3 ) to be “High”, then this illustrates the relative importance of raw material quality. The labour cost ( a1 ) is “Medium” which illustrates that the labour
UNIDENTIFIED RELATIONSHIP TABLE
Attribute
b1 :Payment Delay
Universe
b2 : Cost Control
b3 : Technical Skill
b 4 :Equipment and Infrastructure
Decision Satisfactory Nonbeneficial Satisfactory Beneficial Beneficial Nonbeneficial
x1 x2
Poor Poor
Good Medium
Good Medium
Good Poor
x3 x4 x5 x6
Medium Good Medium Good
Medium Medium Good Poor
Good Good Medium Medium
Good Good Good Poor
TABLE II.
SIMPLIFIED IDENTIFIABLE RELATIONSHIP TABLE
REFERENCES
Conditional Attribute set
Element number
Lower bound
Upper bound
Accuracy of approximation
[1]
Pawlak, Z. Rough Sets[J].International Journal of Computer and Information Sciences, 11, pp. 341㨪356, 1982.
cBeneficial
x 4, x 5
2
2
1.0
[2]
cSatisfactory
x1, x 3
3
3
1.0
cNonbeneficial
x 2, x 6
4
4
1.0
J. Bazan, A. Skowron, and P. Synak, “Discovery of decision rules from experimental data,” Conference Proceedings (RSSC'94) The Third International Workshop on Rough Sets and Soft Computing, San Jose State University, CA, November 10-12, 1994, 526-535. Mentzer, J.T., Dewitt, W., Keebler, J.S., Min, S., Nix, N.W., Smith, C.D., Zacharia, Z.G., 2001. Defining supply chain management. Journal of Business Logistics 22 (2), 1–26. Lambert, D.M., Croxton, K.L., Garcia-Dastugue, S.J., Knemeyer, M., Rogers, D.S., 2006. Supply Chain Management Processes, Partnerships, Performance, 2nd ed. Hartley Press Inc., Jacksonville. Lindqvist, L. J. (1983). Current trends in distribution research in Finland. International Journal of Physical and Logistics Management, 13, 105–116. Greco, S., Matarazzo, B., & Slowinski, R. (2000). Extension of the rough set theory approach to multicriteria decision support. INFOR, 38, 161–196. Daubie, M., Levecq, P., & Meskens, N. (2002). A comparison of the rough sets andrecursive partitioning induction approaches: An application to commercial loans. International Transactions in Operational Research, 9, 681–694. Mickee, T. E. (2003). Rough sets bankruptcy prediction models versus auditor signaling rates. Journal of Forecasting, 22, 569–586. Wei, L.-L., & Zhang, W.-X. (2004). Probabilistic rough sets characterized by fuzzy sets. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 12, 47–60. Kumar, A., Agrawal, D. P., & Joshi, S. D. (2005). Advertising data analysis using rough sets model. International Journal of Information Technology and Decision Making, 4, 263–276. Yan, H., Yu, Z., & Cheng, T. C. E. (2003). A strategic model for supply chain design with logical constraints: Formulation and solution. Computers and Operations Research, 30, 2135–2155. Coleman, J.L. Bhattacharya A.K. and Brace, G. (1995). Supply chain Reengineering: A Supplier Perspective. International Journal of Logistics Management, 6(1), 85-92. Fonsson, P., & Zineldin, M. (2003). Achieving high satisfaction in supplier–dealer working relationships. Supply Chain Management: An International Journal, 8, 224–240.
[3]
[4]
cost is given medium importance. Losses and other costs are “Low”, which shows that the customer wishes to keep the losses and overheads to a minimum. Therefore the decision rules for the above case can be frames as follows
D1 = {a1 = Medium, a 2 = Free, a3 = High, a 4 = Low} V.
CONCLUSION
In this paper we analyzed the importance of supply chain management and the use of rough set theory in optimal cost evaluation for the supply chain of especially small and medium scale enterprises. The proposed model in the paper presents a comprehensive way for knowledge representation and providing precious decision support. The proposed model Takes into account the four major supply chain costs involved in most SME’s and further considers the preferences and requirements of the customer in evaluating the optimal costs.
[5]
[6]
[7]
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